MIT6_851S10_assn08_sol - v i and vertices in B Otherwise we...

6.851 Advanced Data Structures (Spring’10)
Prof. Erik Demaine Dr. Andr´ e Schulz TA: Aleksandar Zlateski
Problem 8 Sample Solutions
Cuckoo Hashing. Consider a cuckoo graph with t edges. We have the probability of any partic-
ular cuckoo graph to be
2
2
t
t
. There are ≤ n
t
m
t
configurations of that graph that contain a cycle
m
(there must be a selection of ≤ n elements with their hash values on the cycle).
�
Hence, the probability that a cuckoo graph with t edges contains a cycle is ≤
n t
m
m
2
t
t
2 t
=
3
1
t
. Since
∞
1
=
1
, we conclude that the probability of a cuckoo graph containing no cycles is at least
t =2
3 t 2
1 / 2.
Conditional Expectations. Consider the following greedy algorithm. We start with two empty
sets A and B . Now we consider all the vertices one by one. For each vertex v
i
we decide to add it
to the set A if there are less edges between v
i
and vertices in A then edges between